Question

For a curve at which the tangent lines at two distinct points coincide, then the curve cannot be

• A. a cubic curve
• B. a quadratic curve
• C. a curve of 4th power
• D. none of these

Answer 1

The correct option is A a cubic curve

Suppose $y=a{x}^3+b{x}^2+cx+d=0(a\ne 0)$ be a cubic curve.
We assume that $(x_1,y_1)$ and $(x_2,y_2)$, $(x_1<x_2)$ are two distinct points on the curve at which tangents coincide.
Then, by Mean value theorem there exists $x_3(x_1<x_3<x_2)$ such that
$\frac{y_2-y_1}{x_2-x_1}=y^{\prime }\left(x_3\right)$
Since, tangent $x_1,x_2,x_3$ are solutions of equation
$3a{x}^2+2bx+c=M$
But, it is a quadratic and thus cannot have more than two roots. Therefore, no such cubic is possible.